Abstract : In the framework of a wavelet analysis, we present an alternative to the minimax approach : the maxiset approach that allows to bring theoretical explanations on some phenomena observed in the practical setting which are not explained by the minimax approach.

In this setting, we aim at investigating the maximal functional spaces (maxisets) where procedures attain a given rate of convergence. In particular, we show that procedures which consist in thresholding coefficients by groups, as vertical-block or horizontal-block thresholding rules, are more efficient in the maxiset sense than procedures which consist in thresholding coefficients individually. Focusing on many large families of shrinkage procedures, we look for optimal procedures in the maxiset sense and we propose a way to build better procedures from optimal procedures for which the maxisets are not embedded.